TRANSLATIVE ISOMORPHISMS AND LEFT-SYMMETRICAL STRUCTURES ON W(M,N)

Let W = W(m, n) be the general Lie algebra of Cartan type and U = U(m, n) the corresponding divided power algebra on which W acts. Translati ve isomorphisms, i.e., isomorphisms of W into W + gl(m, U) of the form A --> A + phi(A), A is an element of W, phi(A) is an element of gl(m, U), are investigated. The mixed product theorems of the author (1986, Sci. Sinica Ser. A 29, No. 6, 570-581) are generalized. Under certain conditions, translative isomorphisms induce left-symmetric structures on W. Classes of left-symmetric structures on W are constructed, amon g which are restricted left-symmetric structures on the restricted Lie algebra W(m, 1). (C) 1996 Academic Press, Inc.